The 28-year-old goalkeeper joined the Red Devils from Atletico Madrid in 2011 and has gone on to make 357 appearances across all competitions.
De Gea has just one year remaining in his existing deal at United and is yet to agree to fresh terms, despite opening negotiations with the club earlier this year.
There has been some suggestion that the club will cash in on the Spaniard to avoid losing him on a free transfer.
But during a press conference in Perth on Wednesday morning, Solskjaer insisted there are “positive” signs that De Gea will ignore transfer rumours and pledge his future to United.
“David has had a summer where he’s been negotiating and we hope we’re going to get it sorted.
“I’m positive, of course. With David I’ve said many times how lucky I am and happy we are to have him in goal,” Solskjaer said.